Theorem dfiun2g 2581

Description: Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44.

Assertion

Ref	Expression
dfiun2g	|- (A.x e. A B e. C -> U_x e. A B = U.{y | E.x e. A y = B})

Distinct variable groups:   y,A   y,B   x,y

Proof of Theorem dfiun2g

Step	Hyp	Ref	Expression
1		hbra1 1684	. . . . . 6 |- (A.x e. A B e. C -> A.xA.x e. A B e. C)
2		ra4 1691	. . . . . . . . 9 |- (A.x e. A B e. C -> (x e. A -> B e. C))
3		clel3g 1888	. . . . . . . . . 10 |- (B e. C -> (w e. B <-> E.z(z = B /\ w e. z)))
4		exancom 1052	. . . . . . . . . 10 |- (E.z(z = B /\ w e. z) <-> E.z(w e. z /\ z = B))
5	3, 4	syl6bb 535	. . . . . . . . 9 |- (B e. C -> (w e. B <-> E.z(w e. z /\ z = B)))
6	2, 5	syl6 22	. . . . . . . 8 |- (A.x e. A B e. C -> (x e. A -> (w e. B <-> E.z(w e. z /\ z = B))))
7	6	pm5.32d 646	. . . . . . 7 |- (A.x e. A B e. C -> ((x e. A /\ w e. B) <-> (x e. A /\ E.z(w e. z /\ z = B))))
8		19.42v 1306	. . . . . . 7 |- (E.z(x e. A /\ (w e. z /\ z = B)) <-> (x e. A /\ E.z(w e. z /\ z = B)))
9	7, 8	syl6bbr 537	. . . . . 6 |- (A.x e. A B e. C -> ((x e. A /\ w e. B) <-> E.z(x e. A /\ (w e. z /\ z = B))))
10	1, 9	exbid 1103	. . . . 5 |- (A.x e. A B e. C -> (E.x(x e. A /\ w e. B) <-> E.xE.z(x e. A /\ (w e. z /\ z = B))))
11		excom 1044	. . . . . 6 |- (E.xE.z(x e. A /\ (w e. z /\ z = B)) <-> E.zE.x(x e. A /\ (w e. z /\ z = B)))
12		19.42v 1306	. . . . . . . 8 |- (E.x(w e. z /\ (x e. A /\ z = B)) <-> (w e. z /\ E.x(x e. A /\ z = B)))
13		an12 484	. . . . . . . . 9 |- ((x e. A /\ (w e. z /\ z = B)) <-> (w e. z /\ (x e. A /\ z = B)))
14	13	exbii 1049	. . . . . . . 8 |- (E.x(x e. A /\ (w e. z /\ z = B)) <-> E.x(w e. z /\ (x e. A /\ z = B)))
15		visset 1809	. . . . . . . . . . 11 |- z e. V
16		eqeq1 1478	. . . . . . . . . . . 12 |- (y = z -> (y = B <-> z = B))
17	16	rexbidv 1661	. . . . . . . . . . 11 |- (y = z -> (E.x e. A y = B <-> E.x e. A z = B))
18	15, 17	elab 1893	. . . . . . . . . 10 |- (z e. {y | E.x e. A y = B} <-> E.x e. A z = B)
19		df-rex 1647	. . . . . . . . . 10 |- (E.x e. A z = B <-> E.x(x e. A /\ z = B))
20	18, 19	bitr 173	. . . . . . . . 9 |- (z e. {y | E.x e. A y = B} <-> E.x(x e. A /\ z = B))
21	20	anbi2i 480	. . . . . . . 8 |- ((w e. z /\ z e. {y | E.x e. A y = B}) <-> (w e. z /\ E.x(x e. A /\ z = B)))
22	12, 14, 21	3bitr4 183	. . . . . . 7 |- (E.x(x e. A /\ (w e. z /\ z = B)) <-> (w e. z /\ z e. {y | E.x e. A y = B}))
23	22	exbii 1049	. . . . . 6 |- (E.zE.x(x e. A /\ (w e. z /\ z = B)) <-> E.z(w e. z /\ z e. {y | E.x e. A y = B}))
24	11, 23	bitr 173	. . . . 5 |- (E.xE.z(x e. A /\ (w e. z /\ z = B)) <-> E.z(w e. z /\ z e. {y | E.x e. A y = B}))
25	10, 24	syl6bb 535	. . . 4 |- (A.x e. A B e. C -> (E.x(x e. A /\ w e. B) <-> E.z(w e. z /\ z e. {y | E.x e. A y = B})))
26		df-rex 1647	. . . 4 |- (E.x e. A w e. B <-> E.x(x e. A /\ w e. B))
27	25, 26	syl5bb 531	. . 3 |- (A.x e. A B e. C -> (E.x e. A w e. B <-> E.z(w e. z /\ z e. {y | E.x e. A y = B})))
28	27	abbidv 1574	. 2 |- (A.x e. A B e. C -> {w | E.x e. A w e. B} = {w | E.z(w e. z /\ z e. {y | E.x e. A y = B})})
29		df-iun 2563	. 2 |- U_x e. A B = {w | E.x e. A w e. B}
30		df-uni 2499	. 2 |- U.{y | E.x e. A y = B} = {w | E.z(w e. z /\ z e. {y | E.x e. A y = B})}
31	28, 29, 30	3eqtr4g 1528	1 |- (A.x e. A B e. C -> U_x e. A B = U.{y | E.x e. A y = B})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  {cab 1461  A.wral 1642  E.wrex 1643  U.cuni 2498  U_ciun 2561

This theorem is referenced by:  dfiun2 2582  iunexg 3853  iunfi 4549  iunopnt 7549  subtop 7596

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rex 1647  df-v 1808  df-uni 2499  df-iun 2563

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